Locally ωα g-Closed Sets in Topological Spaces

In the year 2014, the present authors introduced and studied the concept of ωα g -closed sets in topological spaces. The purpose of this paper to introduce a new class of locally closed sets called ωα g -locally closed sets (briefly lc gωα -sets) and study some of their properties. Also ωα g -locally closed continuous (briefly lc gωα -continuous) functions and its irresolute functions are introduced and studied their properties in topological spaces.


Introduction
The notion of locally closed sets was introduced by Bourbaki [6]. According to him, a subset of a topological space X is locally closed in X if it is the intersection of an open set and closed set in X . Kuratowski and Sierpinski [10] considered the difference of two closed subsets of an ndimensional euclidean space. Implicit in their work is the notion of a locally closed subset of a topological space X . Stone [17] has used the term FG for a locally closed subset as the spaces that in every embedding are locally closed. The results of Borges [5] show that locally closed sets play an important role in the context of simple extension. Ganster and Reilly [8] has introduced locally closed sets, which are weaker forms of both open and closed sets and they used locally closed sets to define LC-continuity and LC-irresoluteness. Sundaram [18] introduced the concept of generalized locally closed sets. After that Balachandran et al. [3], Gnanambal [9], Arokhiarani et al. [1], Pushpalatha [14], Shaik John [15] and P.G. Patil [13] have introduced α -locally closed, generalized locally semi closed, semi generalized locally closed, regular generalized locally closed, strongly locally closed, ω -locally closed and ωα -locally closed sets and their continuous maps in topological spaces respectively. Also various authors have contributed to the development of generalizations of locally closed sets and locally continuous maps in topological spaces.
In this paper, we introduced the notion of ωα g -locally closed sets which are denoted by LC g − ωα sets and study some of the fundamental properties of LC g − ωα sets in generalized topological spaces.

Preliminary
Throughout this paper ) , ( τ X or simply X represents topological space on which no separation axioms are assumed unless and otherwise mentioned. For a subset A of X. cl(A), int(A) and c A denote the closure of A, interior of A and complement of A respectively. If A is a subset of a space τ , then is the smallest τ -closed set containing A and For our analysis, we require the following basic definitions. [12] if A ⊆ int(cl(int(A))).
The complements of the above mentioned sets are called their respective closed sets.
Definition 2.5. A topological space X is said to be a 1. locally closed [8] if where U is open set and V is closed set in X . 2. generalized locally closed (briefly glc -closed) [2] if where U is g-open set and V is g-closed set in X . 3. generalized locally semi-closed (briefly gslc -closed) [9] if where U is g-open set and V is semi-closed set in X . 4. strongly generalized locally closed (briefly lc g * -closed) [14] if where U is strongly g-open set and V is strongly g-closed set in X .

α -locally closed (briefly lc
α -closed) [9] if where U is ω -open set and V is ω -closed set in X . 7. ωα -locally closed (briefly lc ωα -closed [13] If where U is ωα -open set and V is ωα -closed set in X . Definition 2.6. A topological space X is said to be a 1. sub maximal space [7] if every dence subset of X is open in X. 2. door space [2] if every subset of X is either open or closed in X .
is locally closed set in X for locally closed set G of Y .

Locally ωα g -Closed Set
In this section, we introduce lc gωα -sets and study some of their properties.
where U is an open and F is ωα

IJPMS Volume 19
for some open set U. By definition 3.2, . This proves the theorem.

Definition 3.3. Let A be a subset of X . Then A is called ωα
The collection of all ωα g -locally closed sets of X will be denoted by ) ( X LC Gωα .
. Then there exist an open set U and a ωα In a submaximal space X , every ωα g -closed set is gs-closed. Therefore F is gsclosed. Since every open set is gs-open, it follows that A is an intersection of gs-open set U and a gs-closed set F of X. Therefore, . This proves the theorem.

Theorem 3.3. Every locally closed set is ωα
g -locally closed set but not conversely.
Proof. From [4] every closed set is ωα g -closed. Hence the proof follows.
is ωα g -locally closed set but not locally closed set in X .

Remark 3.1. If A is α -locally closed set in X , then A is ωα
g -locally closed in X but not conversely.
Theorem 3.4. For a subset A of X the followings are equivalent.
then there exists a ωα g -open set U and a ωα -closure preserving property. Then the followings are equivalent.
. Then there exist a ωα g -open subset U and a ωα The collection of all * ωα g -locally closed sets of X will be denoted by ) ( * X LC Gωα

Definition 3.5. Let A be a subset of X . Then A is called
Proposition 3.1. For a topological space X the following inclusions hold: Proof. It follows from the fact that every α -closed set is ωα g -closed in X . Remark 3.2. We have the following diagram.
The reverse implications are not true shown in the following example.
. Then we have: (1) The reverse implication need not be true as seen from the above sets.
. Then there exist a ωα g -open set U and a ωα Theorem 3.6. Let X be a ωα g T -space. For a subset A of X the following statements are [4], the intersection of a ωα g -closed set and an α -closed set is ωα Theorem 3.7. For a subset A of X the following statements are equivalent: Proof. Using the fact every α -closed set is ωα g -closed and from theorem 3.6, the proof follows. Proof. Let A be ωα

Proposition 3.5. Every
Proof. Let X be a * ωα g -submaximal space and A be a ωα g -dense subset of X . By proposition 3.4, A is α -dense in X . By assumption, A is ωα g -open and hence X is ωα g -submaximal.
Theorem 3.9. Let X be a Proof.
for some closed set V of X . So V and B are ωα g -closed sets in X . Therefore F is the intersection of ωα g -closed sets V and B. So F is also ωα be a function. Then f is called, . Then there exist a ωα g open set G and a ωα g -closed set H such that